Fixed point theorems for generalized multivalued nonlinear F -contractions

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Research Article

Fixed point theorems for generalized multivalued nonlinear F -contractions

Iram Iqbal^a, Nawab Hussain^b,∗

aDepartment of Mathematics, University of Sargodha, Sargodha, Pakistan.

bDepartment of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.

Communicated by M. Eslamian

Abstract

In this paper, we introduce certain new concepts of α-η-lower semi-continuous and α-η-upper semi- continuous mappings. By using these concepts, we prove some fixed point results for generalized multivalued nonlinear F-contractions in metric spaces and ordered metric spaces. As an application of our results we deduce Suzuki-Wardowski type fixed point results and fixed point results for orbitally lower semi-continuous mappings in complete metric spaces. Our results generalize and extend many recent fixed point theorems including the main results of Minak et al. [G. Minak, M. Olgun, I. Altun, Carpathian J. Math.,31(2015), 241–248], Altun et al. [I. Altun, G. Mınak, M. Olgun, Nonlinear Anal. Model. Control,21(2016), 201–210]

Keywords: α-η-GF-contraction, α-η-F-contraction of Hardy-Rogers type, nonlinearF-contraction.

2010 MSC: 46N40, 47H10, 54H25, 46T99.

1. Introduction and preliminaries

Let (X, d) be a metric space. 2^X denotes the family of all nonempty subsets of X, C(X) denotes the family of all nonempty, closed subsets of X, CB(X) denotes the family of all nonempty, closed, and bounded subsets ofX and K(X) denotes the family of all nonempty compact subsets ofX. It is clear that, K(X)⊆CB(X)⊆C(X)⊆P(X). ForA,B ∈C(X), let

H(A,B) = max (

sup

x∈A

D(x,B),sup

y∈B

D(y,A) )

,

∗Corresponding author

Email addresses: [email protected](Iram Iqbal),[email protected](Nawab Hussain) Received 2016-09-01

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where D(x,B) = inf{d(x, y) :y∈ B}. Then H is called generalized Pompeiu-Hausdorff distance on C(X).

It is well-known thatHis a metric onCB(X), which is called Pompeiu-Hausdorff metric induced byd. For more details see [3],[11].

An interesting generalization of the Banach contraction principle to multivalued mappings is known as Nadler’s fixed point theorem [25]. After this, many authors extended Nadler’s fixed point theorem in many directions (see [10, 12, 24, 29] and references therein). In 2012, Samet et al. [28] defined α-admissible mappings. This notion is generalized by many authors (see [20, 21]). Salimi et al. [27] generalized this idea by introducing the function η and established fixed point theorems. Next, Asl et al. [8] extended these concepts to multivalued mappings by introducing the notion ofα^∗-admissible mappings as follows:

Definition 1.1 ([8]). LetT :X →2^X be a multivalued map on a metric space (X, d), α:X × X →R+ be a function, thenT is anα∗-admissible mapping, if

α(y, z)≥1 implies that α∗(Ty,Tz)≥1, y, z∈ X, where

α∗(A,B) = inf

y∈A,z∈Bα(y, z).

Hussain et al. [19] modified the notion of α∗-admissible as follows:

Definition 1.2([19]). LetT :X →2^X be a multivalued map on a metric space (X, d),α, η:X × X →R+

be two functions whereη is bounded, thenT is anα∗-admissible mapping with respect to η, if α(y, z)≥η(y, z) implies that α∗(Ty,Tz)≥η∗(Ty,Tz), y, z ∈ X,

where

α∗(A,B) = inf

y∈A,z∈Bα(y, z), η∗(A,B) = sup

y∈A,z∈B

η(y, z).

Further, Ali et al. [4] generalized Definition 1.2 in the following way.

Definition 1.3 ([4]). LetT :X →2^X be a multivalued map on a metric space (X, d),α, η:X×X →R+

be two functions. We say thatT is generalizedα∗-admissible mapping with respect toη, if α(y, z)≥η(y, z) implies that α(u, v)≥η(u, v), for all u∈Ty, v∈Tz.

In 2014, Hussain et al. [16] introduced the notion of α-η continuous mappings as follows:

Definition 1.4 ([16]). Let (X, d) be a metric space, α, η :X × X → [0,∞) and T :X → X be functions.

Then T is anα-η-continuous mapping onX, if for given z∈X and sequence{z_n}with zn→z as n→ ∞, α(zn, zn+1)≥η(zn, zn+1), for all n∈N⇒ Tzn→ Tz.

After that Hussain et al. [15] generalized Definition 1.4 to multivalued maps.

Definition 1.5([15]). LetT :X →2^X be a multivalued map on a metric space (X, d),α, η:X × X →R+

be two functions. We say thatT is α-η continuous multivalued mapping, if for given z ∈X and sequence {z_n} with z_n → z as n → ∞, α(z_n, z_n+1) ≥ η(z_n, z_n+1), for all n ∈ N we have Tz_n → Tz. That is, limn→∞d(zn, z) = 0 andα(zn, zn+1)≥η(zn, zn+1) implies limn→∞H(Tzn,Tz) = 0.

Recently, Wardowski [31] defined F-contraction and proved a fixed point result as a generalization of the Banach contraction principle for this contraction. This idea has been extended in many directions (see [1, 14, 17] and references therein). Hussain et al. [18] broadened this idea toα-GF-contraction with respect to a general family of functionsG. Following Wardowski and Hussain, we denote byF, the set of all functions F :R⁺→Rsatisfying the following conditions:

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(F₁) F is strictly increasing;

(F₂) for all sequence{α_n} ⊆R⁺, limn→∞αn= 0, if and only if limn→∞F(α_n) =−∞;

(F₃) there exists 0< k <1 such that lim_α→0⁺α^kF(α) = 0, F_∗, ifF also satisfies the following:

(F₄) F(infA) = infF(A) for all A⊂(0,∞) with infA >0, G, the set of all functionsG :R⁺

4 →R⁺ satisfying:

(G) for allt₁, t_t, t₃, t₄ ∈R⁺ with t₁t₂t₃t₄= 0 there exists τ >0 such thatG(t₁, t₂, t₃, t₄) =τ.

On unifying the concepts of Wardowski’s and Nadlers, Altun et al. [5] gave the concept of multivalued F-contractions and established some fixed point results. On the other side, Minak et al. [23], extended the results of Wardowski as follows:

Theorem 1.6 ([23]). Let (X, d) be a complete metric space, T : X → K(X) and F ∈ F. If there exists τ >0 such that for any z∈ X with d(z,Tz)>0, there existsy∈ F_σ^z satisfying

τ+F(D(y,Ty))≤ F(d(z, y)), where

F_σ^z ={y∈ Tz:F(d(z, y))≤ F(D(z,Tz)) +σ},

thenT has a fixed point in X providedσ < τ and z→d(z,Tz) is lower semi-continuous.

Theorem 1.7 ([23]). Let (X, d) be a complete metric space, T : X → C(X) and F ∈ F_∗. If there exists τ >0 such that for any z∈ X with d(z,Tz)>0, there existsy∈ F_σ^z satisfying

τ+F(D(y,Ty))≤ F(d(z, y)),

thenT has a fixed point in X providedσ < τ and z→d(z,Tz) is lower semi-continuous.

Minak et al. [23] also showed that F_σ^z 6= ∅ in both cases when F ∈ F and F ∈ F_∗. The aim of the present paper is to introduce the concept of α-η-semicontinuous multivalued mappings and to prove fixed point theorem for multivalued nonlinearF-contractions that generalize the results of Altun et al. [6], Minak et al. [23], Olgun et al. [26] and Hussain et al. [18]. The following lemmas will be used in the sequel.

Lemma 1.8 ([3]). Let T :X → Y be a multivalued function, then the following statements are equivalent.

1. T is lower semi-continuous.

2. V ⊂ Y ⇒ T⁻¹[int(V)] is open in X, where int(V) denotes the interior of V.

Lemma 1.9 ([3]). Let T :X → Y be a multivalued function, then the following statements are equivalent.

1. T is upper semi-continuous.

2. V ⊂ Y ⇒ T⁻¹[V] is closed in X, where V denotes the closure ofV.

2. Fixed point results for modified α-η-GF-contraction We begin this section with the following definitions.

Definition 2.1. Let T :X →2^X be a multivalued map on a metric space (X, d), α, η : X × X →R+ be

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two functions. We say thatT is α-η lower semi-continuous multivalued mapping on X, if for given z∈X and sequence{z_n}with

n→∞lim d(z_n, z) = 0, α(z_n, z_n+1)≥η(z_n, z_n+1), for all n∈N, implies

n→∞lim infD(z_n,Tz_n)≥D(z,Tz).

Definition 2.2. Let T :X →2^X be a multivalued map on a metric space (X, d), α, η : X × X →R+ be two functions. We say thatT is α-η upper semi-continuous multivalued mapping onX, if for given z∈X and sequence{z_n}with

n→∞lim supD(z_n,Tz_n)≤D(z,Tz).

Lemma 2.3. Let T : X → 2^X be a multivalued map on a metric space (X, d), α, η : X × X → R+ be two functions. Then T is α-η continuous, if and only if it is α-η upper semi-continuous and α-η lower semi-continuous.

Proof. Suppose that T is α-η upper semi-continuous and α-η lower semi-continuous. Then there exists a sequence {z_n}in X and z∈ X with

n→∞lim infD(zn,Tzn)≥D(z,Tz), (2.1) and

n→∞lim supD(zn,Tzn)≤D(z,Tz). (2.2) From (2.1) and (2.2), we get that D(zn,Tzn) → D(z,Tz) as n → ∞. This is possible only when Tz_n→ Tz. Consequently,T isα-η continuous.

Conversely, suppose that T is α-η continuous. Then there exists a sequence {z_n} in X and z ∈ X with zn → z as n → ∞ and α(zn, zn+1) ≥ η(zn, zn+1) for all n ∈ N implies Tzn → Tz as n → ∞. This implies that D(z_n,Tz_n) → D(z,Tz) as n → ∞ or limn→∞D(z_n,Tz_n) = D(z,Tz). From here it follows that limn→∞infD(z_n,Tz_n) ≥ D(z,Tz) and limn→∞supD(z_n,Tz_n) ≤ D(z,Tz). Hence T is α-η upper semi-continuous andα-η lower semi-continuous.

Remark 2.4. As semi-continuity is a weaker property than continuity, anα-ηupper semi-continuous andα-η lower semi-continuous mapping need not to be α-η continuous mapping, as shown in the examples below.

Example 2.5. Let X = R with usual metric d. Then (X, d) is a metric space. Define T₁ : X → 2^X, α, η:X × X →R+ by

T₁z=

{0} if z6= 0, [−1,1] if z= 0, α(y, z) =

1 if z, y6= 0, 0 if z=y= 0, and η(z, y) = ¹₂, for all z, y∈ X.

Firstly, we show that T₁ is not lower semi-continuous multivalued map. For this, let V = [−1,1]⊂2^X,

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then T₁⁻¹(int(V)) =T₁⁻¹((−1,1)) = {0} which is not open in R, so by Lemma 1.8, T₁ is not lower semi- continuous. But T₁ is α-η lower semi-continuous multivalued map. Indeed, α(zn, zn+1) ≥ η(zn, zn+1) for sequence zn of non-zero real numbers. Here arises two cases:

Case I.zn→z= 0.

If zn → 0, then T₁zn = {0} and T₁z = [−1,1] such that D(zn,T₁zn) = D(zn,{0}) = zn and D(z,T₁z) = D(0,[−1,1]) = 0. This implies that

n→∞lim infD(z_n,Tz_n) = lim

n→∞infz_n=z= 0 =D(z,Tz).

Case II.zn→z6= 0.

Ifz_n→z, thenT₁z_n={0}andT₁z={0}such thatD(z_n,T₁z_n) =D(z_n,{0}) =z_n andD(z,T₁z) =z. This implies that

n→∞lim infD(zn,T₁zn) = lim

n→∞infzn=z=D(z,T₁z).

On the other hand, in Case I we have

n→∞lim H(T₁z_n,T₁z) = 1.

Hence T₁ is not α-η-continuous multivalued map.

Example 2.6. Consider X the same as in Example 2.5. Define T₂:X →2^X,α, η:X × X →R+ by T₂z=

[−1,1] if z6= 0, {0} if z= 0, α(z, y) =

0 if z, y6= 0, 2 if z=y= 0, and η(z, y) = ¹₄, for all z, y∈ X.

Firstly, we show that T₂ is not upper semi-continuous multivalued map. For this, letV = [−1,1]⊂2^X, thenT₂⁻¹(V) =T₂⁻¹([−1,1]) =R\{0}= (−∞,0)∪(0,∞), which is not closed inR, so by Lemma 1.9, T₂ is not upper semi-continuous. But T₂ is α-η upper semi-continuous multivalued map. Indeed, α(zn, zn+1) ≥ η(zn, zn+1) for sequence zn= 0 for alln∈N. Thenzn approaches toz= 0 only. Therefore, If zn→0, then T₂z_n={0} andT₂z={0}. This implies that

n→∞lim supD(z_n,T₂z_n) = 0 =D(z,T₂z).

On the other hand,

n→∞lim H(T₂z_n,T₂z) = 1.

Hence T₂ is not α-η-continuous multivalued map.

Remark 2.7. Let T :X →2^X be a multivalued map on a metric space (X, d). Let f :X →R, defined by f(z) =D(z,Tz), for allz∈ X, be a lower semi-continuous mapping. Takeα(z, y) =η(z, y), for allz, y∈ X, then forz∈ X and a sequence{z_n} with

n→∞lim d(z_n, z) = 0, α(z_n, z_n+1)≥η(z_n, z_n+1) for alln∈N, we have

n→∞lim inff(z_n)≥f(z), and so

n→∞lim infD(z_n,Tz_n)≥D(z,Tz).

This shows thatT isα-ηlower semi-continuous mapping. But ifT isα-ηlower semi-continuous mapping, thenf needs to be lower semi-continuous as shown in Example 2.12. Similarly, if f :X →Ris upper semi- continuous mapping then,T is α-η upper semi-continuous mapping but not conversely.

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Theorem 2.8. Let (X, d) be a complete metric space and α, η : X × X → R+ be two functions. Let T :X →K(X), F ∈F and G ∈G fulfilling the following assertions:

(1) if for anyz∈ X with D(z,Tz)>0, there existsy∈ F_σ^z with α(z, y)≥η(z, y) satisfying G(D(z,Tz), D(y,Ty), D(z,Ty), D(y,Tz)) +F(D(y,Ty))≤ F(d(z, y));

(2) T is generalizedα∗-admissible mapping with respect to η;

(3) T isα-η lower semi-continuous mapping;

(4) there existsz₀ ∈ X and y₀∈ Tz₀ such that α(z₀, y₀)≥η(z₀, y₀).

ThenT has a fixed point in X provided σ < τ.

Proof. Let z0 ∈ X, since Tz ∈ K(X) for every z ∈ X, the set F_σ^z is non-empty for any σ > 0, then there exists z1 ∈ F_σ^z⁰ and by hypothesis α(z0, z1) ≥ η(z0, z1). Assume that z1 ∈ T/ z1, otherwise z1 is the fixed point ofT. Then, sinceTz₁ is closed,D(z₁,Tz₁)>0, so from condition (1), we have

G(D(z₀,Tz₀), D(z₁,Tz₁), D(z₀,Tz₁), D(z₁,Tz₀)) +F(D(z₁,Tz₁))≤ F(d(z₀, z₁)). (2.3) Now for z₁ ∈ X there exists z₂ ∈ F_σ^z¹ with z₂ ∈ T/ z₂, otherwise z₂ is the fixed point of T, since Tz₂ is closed, so, D(z₂,Tz₂) > 0. Since T is generalized α∗-admissible mapping with respect to η, then α(z1, z2)≥η(z1, z2). Again by using condition (1), we get

G(D(z₁,Tz1), D(z2,Tz2), D(z1,Tz2), D(z2,Tz1)) +F(D(z₂,Tz2))≤ F(d(z₁, z2)).

On continuing recursively, we get a sequence {z_n}_n∈_N in X such that zn+1 ∈ F_σ^zⁿ, zn+1 ∈ T/ zn+1, α(z_n, z_n+1)≥η(z_n, z_n+1) and

G(D(z_n,Tz_n), D(z_n+1,Tz_n+1), D(z_n,Tz_n+1), D(z_n+1,Tz_n)) +F(D(z_n+1,Tz_n+1))≤ F(d(z_n, z_n+1)).

Asz_n+1∈ Tz_n, this implies that

G(D(z_n,Tz_n), D(z_n+1,Tz_n+1), D(z_n,Tz_n+1),0) +F(D(z_n+1,Tz_n+1))≤ F(d(z_n, z_n+1)). (2.4) From (G) there exists τ >0 such that

G(D(z_n,Tz_n), D(z_n+1,Tz_n+1), D(z_n,Tz_n+1),0) =τ.

From equation (2.4), we get that

F(D(z_n+1,Tz_n+1))≤ F(d(z_n, z_n+1))−τ. (2.5) Since z_n+1 ∈ F_σ^zⁿ, we have

F(d(z_n, z_n+1))≤ F(D(z_n,Tz_n)) +σ. (2.6) Combining equations (2.5) and (2.6) gives

F(D(z_n+1,Tz_n+1))≤ F(D(z_n,Tz_n)) +σ−τ. (2.7) Since Tzn and Tzn+1 is compact, there exists zn+1 ∈ Tzn and zn+2 ∈ Tzn+1 such that d(zn, zn+1) = D(z_n,Tz_n) andd(z_n+1, z_n+2) =D(z_n+1,Tz_n+1), so equation (2.7) implies

F(d(z_n+1, z_n+2))≤ F(d(z_n, z_n+1)) +σ−τ. (2.8)

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By using equation (2.8), we get

F(d(zn+1, zn+2))≤ F(d(zn, zn+1)) +σ−τ

≤ F(d(zn−1, z_n)) + 2σ−2τ ...

≤ F(d(z0, z1)) +nσ−nτ

=F(d(z0, z1))−n(τ −σ).

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By letting limit as n → ∞ in equation (2.9), we get limn→∞F(d(z_n+1, zn+2)) = −∞, so by (F2), we obtain

n→∞lim d(zn+1, zn+2) = 0. (2.10)

Now from (F3), there exists 0< k <1 such that

n→∞lim[d(z_n+1, z_n+2)]^kF(d(z_n+1, z_n+2)) = 0. (2.11) By equation (2.9), we get

n→∞lim[d(z_n+1, z_n+2)]^k[F(d(z_n+1, z_n+2))−d(z₀, z₁)]≤ −n(τ −σ)[d(z_n+1, z_n+2)]^k ≤0. (2.12) By taking limit as n→ ∞ in equation (2.12) and applying equations (2.10) and (2.11), we have

n→∞lim n[d(z_n+1, z_n+2)]^k = 0.

This implies that there exists n₁ ∈ N such that n[d(z_n+1, z_n+2)]^k ≤ 1, or d(z_n+1, z_n+2) ≤ ¹

n^1/k, for all n > n₁. Next, for m > n > n₁ we have

d(z_n, z_m)≤

m−1

X

i=n

d(z_i, z_i+1)≤

m−1

X

i=n

1 i^1/k, since 0 < k < 1, Pm−1

i=n 1

i^1/k converges. Therefore, d(zn, zm) → 0 as m, n → ∞. Thus, {z_n} is a Cauchy sequence. SinceX is complete, there existsz^∗ ∈ X such thatz_n→z^∗ asn→ ∞. From equations (2.7) and (2.10), we have

n→∞lim D(zn,Tzn) = 0.

Since T is α-η lower semi-continuous mapping, then 0≤D(z, T z)≤ lim

n→∞infD(zn,Tzn) = 0.

Thus,T has a fixed point.

Theorem 2.9. Let (X, d) be a complete metric space and α, η : X × X → R+ be two functions. Let T :X →C(X), F ∈F_∗ and G ∈G satisfy all assertions of Theorem2.8. Then T has a fixed point in X. Proof. Let z₀ ∈ X, since Tz∈C(X) for every z∈ X and F ∈F_∗, the set F_σ^z is non-empty for anyσ >0, then there exists z₁ ∈ F_σ^z⁰ and by hypothesis α(z₀, z₁) ≥ η(z₀, z₁). Assume that z₁ ∈ T/ z₁, otherwise z₁ is the fixed point of T. Then, sinceTz1 is closed, D(z1,Tz1) >0, so from condition (1) of Theorem 2.8, we have

G(D(z₀,Tz₀), D(z₁,Tz₁), D(z₀,Tz₁), D(z₁,Tz₀)) +F(D(z₁,Tz₁))≤ F(d(z₀, z₁)).

Now for z₁ ∈ X there exists z₂ ∈ F_σ^z¹ with z₂ ∈ T/ z₂, otherwise z₂ is the fixed point of T, since

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Tz₂ is closed, so, D(z₂,Tz₂) > 0. Since T is generalized α∗-admissible mapping with respect to η, then α(z1, z2)≥η(z1, z2). Again by using condition (1) of Theorem 2.8, we get

G(D(z₁,Tz1), D(z2,Tz2), D(z1,Tz2), D(z2,Tz1)) +F(D(z₂,Tz2))≤ F(d(z₁, z2)).

On continuing recursively, we get a sequence {z_n}_n∈_N in X such that zn+1 ∈ F_σ^zⁿ, zn+1 ∈ T/ zn+1, α(zn, zn+1)≥η(zn, zn+1) and

G(D(z_n,Tz_n), D(z_n+1,Tz_n+1), D(z_n,Tz_n+1), D(z_n+1,Tz_n)) +F(D(z_n+1,Tz_n+1))

≤ F(d(z_n, z_n+1)).

The rest of the proof can be completed as the proof of Theorem 2.8.

Corollary 2.10. Let (X, d) be a complete metric space and α, η : X × X → R+ be two functions. Let T : X → K(X) and F ∈ F fulfill the conditions (2)-(4) of Theorem 2.8 and if for any z ∈ X with D(z,Tz)>0, there existsy∈ F_σ^z with α(z, y)≥η(z, y) satisfying

τ+F(D(y,Ty))≤ F(d(z, y)), thenT has a fixed point in X providedσ < τ.

Proof. Define G_L:R⁺

4 →R⁺ by G(t₁, t₂, t₃, t₄) =Lmin{t₁, t₂, t₃, t₄}+τ, where L∈R⁺ and τ >0. Then G_L∈G(see Example 2.1 of [18]). Therefore, the result follows by taking G=G_L in Theorem 2.8.

Corollary 2.11. Let (X, d) be a complete metric space and α, η : X × X → R+ be two functions. Let T :X →C(X) andF ∈F_∗ satisfy all conditions of Corollary2.10. Then T has a fixed point in X.

Proof. By defining same G_L as in Corollary 2.10 and using Theorem 2.9, we get the required result.

Example 2.12. LetX = ₁

2ⁿ⁻¹ :n∈N ∪ {0}with usual metric d. Then (X, d) is a metric space. Define T :X →K(X),α, η:X × X →R+,G:R⁴ →R⁺ andF :R⁺ →R by

Tz= ₁

2ⁿ if z= ₂n−1¹ , {0} if z= 0, α(z, y) =

2 if z= ₂n−1¹ ,

1

2 if z= 0,

η(z, y) = 1, for allz, y∈ X,G(t₁, t₂, t₃, t₄) =τ, whereτ >0 andF(r) = ln(r). Then D(z,Tz) =

₁

2ⁿ if z= ₂n−1¹ , 0 if z= 0.

Let D(z,Tz)>0, then z= ₂n−1¹ , so, Tz=₁

2ⁿ . Thus fory= ₂¹n ∈ Tz, we have F(d(z, y))− F(D(z,Tz)) =F

1 2ⁿ

− F 1

2ⁿ

= 0.

Therefore, y∈ F_σ^z forσ >0 with α(z, y)≥η(z, y) and F(D(y,Ty))− F(d(z, y)) =F

1 2ⁿ⁺¹

− F 1

2ⁿ

= ln 1

2ⁿ⁺¹

−ln 1

2ⁿ

= ln 2ⁿ

2ⁿ⁺¹

= ln 1

2

=−ln 2.

Hence τ+F(D(y,Ty))≤ F(d(z, y)) is satisfied for 0< σ < τ ≤ln 2.

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Since α(z, y) ≥ η(z, y) when z, y ∈ ₁

2ⁿ⁻¹ :n∈N , this implies that α(u, v) = 2 > 1 = η(u, v) for all u∈ Tz and v∈ Ty. HenceT is generalizedα∗-admissible mapping with respect toη.

Next, let limn→∞d(zn, z) = 0 and α(zn, zn+1) ≥ η(zn, zn+1), for all n ∈ N, then zn ∈ ₁

2ⁿ⁻¹ :n∈N . This implies thatTz_n=₁

2ⁿ and D(z_n,Tz_n) = ₂¹n, for alln∈N. Here arises two cases:

Case I.zn→z= 0.

Then Tz={0} and D(z,Tz) = 0. Thus

n→∞lim infD(zn,Tzn) = lim

n→∞inf 1

2ⁿ

≥0 =D(z,Tz).

Case II.zn→z= ₂n−1¹ . Then Tz=₁

2ⁿ and D(z,Tz) = ₂¹n. Thus

n→∞inf( 1 2ⁿ)

= 1

2ⁿ =D(z,Tz).

Hence T is α-η lower semi-continuous mapping. Thus, all conditions of Corollary 2.10 (and Theorem 2.8) hold and 0 is a fixed point ofT.

On the other hand, define f :X →R, by f(z) =D(z,Tz), for allz∈ X. Then

z→1liminff(z) = 0 1

2 =f(1).

Hence f is not lower semi-continuous mapping at z = 1. That is, Theorems 1.6 and 1.7 can not be applied for this example.

Example 2.13. Consider the sequence{S_n}_n∈_N as follows:

S1= 1, S₂= 1 + 2,

...

Sn= 1 + 2 + 3 +...+n= n(n+ 1)

2 ,

...

Let X = {S_n:n∈N} with usual metric d. Then (X, d) is a metric space. Define T : X → K(X), α, η:X × X →R+,G:R⁴ →R⁺ andF :R⁺→R by

Tz=

{S_n−1, Sn+1} if z=Sn, n >2,

{z} otherwise,

α(z, y) =

3 if z∈ {S_n:n≥2}, 1 otherwiswe,

η(z, y) = 2, for all z, y ∈ X, G(t₁, t₂, t₃, t₄) = Lmin{t₁, t₂, t₃, t₄}+τ, where τ = _e¹n, n ∈ N, L ∈ R⁺ and F(r) = ln(r). Then

D(z,Tz) =

|n| if z=S_n, n >2, 0 otherwise.

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Let D(z,Tz)>0, then z=S_n, n >2, so, Tz={S_n−1, S_n+1}. Thus fory =Sn−1∈ Tz, we have F(d(z, y))− F(D(z,Tz)) =F(|n|)− F(|n|) = 0.

Therefore, y∈ F_σ^z forσ= _en+1¹ ,n∈Nwith α(z, y)≥η(z, y) and

F(D(y,Ty))− F(d(z, y)) =F(|n−1|)− F(|n|)

= ln (|n−1|)−ln (|n|)

= ln

|n−1|

|n|

<−1 eⁿ.

This implies that τ+F(D(y,Ty))≤ F(d(z, y)). SinceD(z,Ty) = 0, we have,

G(D(z,Tz), D(y,Ty), D(z,Ty), D(y,Tz)) +F(D(y,Ty)) =τ +F(D(y,Ty))

≤ F(d(z, y)).

Hence condition (1) of Theorem 2.8 is satisfied for 0< σ= _en+1¹ < τ = _e¹n.

Since α(z, y) ≥ η(z, y) when z, y ∈ {S_n:n≥2}, this implies that α(u, v) = 3 > 2 = η(u, v) for all u∈ Tz and v∈ Ty. HenceT is generalizedα∗-admissible mapping with respect toη.

Next, let limn→∞d(z_n, z) = 0 andα(z_n, z_n+1)≥η(z_n, z_n+1), for alln∈N, thenz_n∈ {S_n:n∈N, n≥2}.

Here arises two cases:

Case I.z_n∈ {S_n:n >2}.

Then Tz_n={S_n−1, S_n+1} and D(z_n,Tz_n) =|n|, for all n∈N. Subcase I.zn→z=Sn, n >2.

Then Tz={S_n−1, Sn+1}and D(z,Tz) =|n|. Thus

n→∞inf (|n|)

=|n|=D(z,Tz).

Subcase II.zn→z=S1.

Then Tz={S₁} andD(z,Tz) = 0. Thus

n→∞inf (|n|)

≥0 =D(z,Tz).

Subcase III.z_n→z=S₂.

Then Tz={S₂} andD(z,Tz) = 0. Thus

n→∞lim infD(zn,Tzn) = lim

n→∞inf (|n|)

≥0 =D(z,Tz).

Case II.z_n∈ {S₂}.

Then zn approaches toS2 only. Therefore,Tzn={z_n}and Tz={z}. This implies that

n→∞lim infD(z_n,Tz_n) = 0 =D(z,Tz).

Hence T is α-η lower semi-continuous mapping. Thus, all the conditions of Theorem 2.8 hold and {S₁, S₂} is set of fixed points ofT.

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As an application of Theorems 2.8 and 2.9, we get the following results.

Theorem 2.14. Let (X, d) be a complete metric space and α, η : X × X → R+ be two functions. Let T :X →K(X), F ∈F andG ∈Gfulfill the conditions (2) and (4) of Theorem 2.8. If for anyy, z ∈ X with α(z, y)≥η(z, y) and H(Tz,Ty)>0 we have

G(D(z,Tz), D(y,Ty), D(z,Ty), D(y,Tz)) +F(H(Tz,Ty))≤ F(d(z, y)), thenT has a fixed point inX provided T is α-η continuous mapping.

Proof. By Lemma 2.3, we have T is α-η-lower semi-continuous mapping. Also, forz ∈ X and y∈ F_σ^z with D(z,Tz)>0 we have

G(D(z,Tz), D(y,Ty), D(z,Ty), D(y,Tz)) +F(D(y,Ty))

≤ G(D(z,Tz), D(y,Ty), D(z,Ty), D(y,Tz)) +F(H(Tz,Ty))

≤ F(d(z, y)).

Thus, all the conditions of Theorem 2.8 are satisfied, so, T has a fixed point.

By similar arguments of Theorem 2.14, we state the following and omit its proof.

Theorem 2.15. Let (X, d) be a complete metric space and α, η : X × X → R+ be two functions. Let T :X →C(X), F ∈F_∗ and G ∈G satisfy all assertions of Theorem2.14. Then T has a fixed point inX.

On consideringG=G_L, as in Corollary 2.10, Theorems 2.14 and 2.15 reduce to the following corollaries.

Corollary 2.16. Let (X, d) be a complete metric space and α, η : X × X → R+ be two functions. Let T : X → K(X) and F ∈ F fulfill the conditions (2) and (4) of Theorem 2.8. If for any y, z ∈ X with α(z, y)≥η(z, y) and H(Tz,Ty)>0 we have

τ +F(H(Tz,Ty))≤ F(d(z, y)), thenT has a fixed point inX provided T is α-η continuous mapping.

Corollary 2.17. Let (X, d) be a complete metric space and α, η : X × X → R+ be two functions. Let T :X →C(X) andF ∈F∗ satisfy all assertions of Corollary 2.16. Then T has a fixed point in X.

Theorem 2.18. Let (X, d) be a complete metric space, T :X → K(X), F ∈ F and G ∈ G. If for z ∈ X withD(z,Tz)>0, there exists y∈ F_σ^z satisfying

G(D(z,Tz), D(y,Ty), D(z,Ty), D(y,Tz)) +F(D(y,Ty))≤ F(d(z, y)), thenT has a fixed point in X providedσ < τ and z→D(z,Tz) is lower semi-continuous.

Proof. Define α(z, y) =d(z, y) = η(z, y) for all z, y ∈ X. Then α(u, v) = d(z, y) =η(u, v), for all u ∈ Tz and v ∈ Ty, that is, T is generalized α∗-admissible mapping with respect to η. Since z → D(z,Tz) is lower semi-continuous, therefore by Remark 2.7, T is α-η-lower semi-continuous. Thus, all the conditions of Theorem 2.8 holds. Hence T has a fixed point inX.

Theorem 2.19. Let (X, d) be a complete metric space, T :X →C(X), F ∈F_∗ and G ∈G. If for z ∈ X withD(z,Tz)>0, there exists y∈ F_σ^z satisfying

G(D(z,Tz), D(y,Ty), D(z,Ty), D(y,Tz)) +F(D(y,Ty))≤ F(d(z, y)), thenT has a fixed point in X providedσ < τ and z→D(z,Tz) is lower semi-continuous.

Proof. By defining α(z, y) and η(z, y) the same as in proof of Theorem 2.18 and by using Theorem 2.8, we get the required result.

Remark 2.20. By takingG =G_L, as in Corollary 2.11, in Theorems 2.18 and 2.19, we get Theorems 1.6 and 1.7.

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3. Fixed point results for α-η-F-contraction of Hardy-Rogers type

In this section we establish certain fixed point results for α-η-F-contraction of Hardy-Rogers type.

Theorem 3.1. Let (X, d) be a complete metric space and α, η : X × X → R+ be two functions. Let T :X →K(X) and F ∈F fulfill the following assertions:

1. T is generalizedα∗-admissible mapping with respect to η;

2. T isα-η lower semi-continuous mapping;

3. there exist z₀ ∈ X and y₀ ∈ Tz₀ such thatα(z₀, y₀)≥η(z₀, y₀);

4. there exist σ >0 and a function τ : (0,∞)→(σ,∞) such that

t→slim⁺infτ(t)> σ, for alls≥0,

and for any z∈ X with D(z,Tz)>0, there existsy∈ F_σ^z with α(z, y)≥η(z, y) satisfying τ(d(z, y)) +F(D(y,Ty))≤ F(a₁d(z, y) +a₂D(z,Tz) +a₃D(y,Ty)

+a₄D(z,Ty) +a₅D(y,Tz)), where a₁, a₂, a₃, a₄, a₅∈[0,+∞) such that a₁+a₂+a₃+ 2a₄= 1 and a₃ 6= 1.

ThenT has a fixed point in X.

Proof. Let z₀ ∈ X, since Tz ∈ K(X) for every z ∈ X, the set F_σ^z is non-empty for any σ > 0, then there exists z1 ∈ F_σ^z⁰ and by hypothesis α(z0, z1) ≥ η(z0, z1). Assume that z1 ∈ T/ z1, otherwise z1 is the fixed point ofT. Then, sinceTz1 is closed,D(z1,Tz1)>0, so, from (4), we have

τ(d(z0, z1)) +F(D(z1,Tz1))≤ F(a₁d(z0, z1) +a2D(z0,Tz0) +a3D(z1,Tz1) +a4D(z0,Tz1) +a5D(z1,Tz0)).

Now for z1 ∈ X there exists z2 ∈ F_σ^z¹ with z2 ∈ T/ z2, otherwise z2 is the fixed point of T, since Tz2 is closed, so, D(z2,Tz2) > 0. Since T is generalized α∗-admissible mapping with respect to η, then α(z₁, z₂)≥η(z₁, z₂). Again by using (4), we get

τ(d(z₁, z₂)) +F(D(z₂,Tz₂))≤ F(a₁d(z₁, z₂) +a₂D(z₁,Tz₁) +a₃D(z₂,Tz₂) +a4D(z1,Tz2) +a5D(z2,Tz1)).

On continuing recursively, we get a sequence {z_n}_n∈_N in X such that zn+1 ∈ F_σ^zⁿ, zn+1 ∈ T/ zn+1, α(z_n, z_n+1)≥η(z_n, z_n+1) and

τ(d(z_n, z_n+1)) +F(D(z_n+1,Tz_n+1))≤ F(a₁d(z_n, z_n+1) +a₂D(z_n,Tz_n) +a₃D(z_n+1,Tz_n+1) +a4D(zn,Tzn+1) +a5D(zn+1,Tzn)).

Aszn+1∈ Tzn, this implies that

τ(d(z_n, z_n+1)) +F(D(z_n+1,Tz_n+1))≤ F(a₁d(z_n, z_n+1) +a₂D(z_n,Tz_n)

+a₃D(z_n+1,Tz_n+1) +a₄D(z_n,Tz_n+1)). (3.1) Since z_n+1 ∈ F_σ^zⁿ, we have

F(d(z_n, zn+1))≤ F(D(zn,Tzn)) +σ. (3.2) As Tz_n and Tz_n+1 is compact, there exist z_n+1 ∈ Tz_n and z_n+2 ∈ Tz_n+1 such that d(z_n, z_n+1) = D(zn,Tzn) andd(zn+1, zn+2) =D(zn+1,Tzn+1), so equations (3.1) and (3.2) imply

τ(d(zn, zn+1)) +F(d(zn+1, zn+2))≤ F(a1d(zn, zn+1) +a2d(zn, zn+1) +a₃d(z_n+1, z_n+2) +a₄d(z_n, z_n+2)),

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and

F(d(z_n, zn+1))≤ F(d(zn, zn+1)) +σ. (3.3) Let d_n=d(z_n, z_n+1), forn∈N, then

τ(d_n) +F(d_n+1)≤ F((a₁+a₂)d_n+a₃d_n+1+a₄d(z_n, z_n+2)

≤ F((a1+a2+a4)dn+ (a3+a4)dn+1). (3.4) Assume that there existsn∈Nsuch that dn+1 ≥dn, then from (3.4), we get

τ(dn) +F(dn+1)≤ F(dn+1).

This is a contradiction to the fact that τ(dn) > 0. Hence dn+1 < dn for all n ∈ N. This shows that sequence {d_n} is decreasing. Therefore, there exists δ ≥0 such that limn→∞d_n=δ. Now let δ >0. From (3.4), we get

τ(dn) +F(dn+1)≤ F(dn). (3.5)

Combining (3.3) and (3.5) gives

F(d_n+1)≤ F(d_n) +σ−τ(d_n)

≤ F(dn−1) + 2σ−τ(d_n)−τ(dn−1) ...

≤ F(d0) +nσ−τ(dn)−τ(dn−1)− · · · −τ(d0).

(3.6)

Let τ(d_p_n) = min{τ(d₀), τ(d₁),· · · , τ(d_n)}for all n∈N. From (3.6), we get

F(d_n+1)≤ F(d0) +n(σ−τ(dpn)). (3.7) From (3.6), we also get

F(D(zn+1,Tzn+1))≤ F(D(z₀,Tz0)) +n(σ−τ(dpn)).

Now consider the sequence{τ(d_p_n)}. We distinguish two cases.

Case 1. For each n∈N, there ism > n such that τ(dpn) > τ(dpm). Then we obtain a subsequence {d_p_nk} of{d_p_n} withτ(d_p_nk)> τ(d_p_nk+1) for all k. Sinced_p_nk →δ⁺, we deduce that

k→∞lim infτ(d_p_nk)> σ.

HenceF(d_n_k)≤ F(d₀) +n(σ−τ(d_p_nk)) for allk. Consequently, limk→∞F(d_n_k) =−∞and by (F2), we obtain limk→∞dp_nk = 0, which contradicts that limn→∞dn>0.

Case 2. There isn0∈Nsuch thatτ(dpn0)> τ(dpm) for allm > n0. Then F(d_m)≤ F(d₀) +m(σ−τ(dpn0)) for allm > n0. Hence limm→∞F(d_m) =−∞, so lim_m→∞dm = 0, which contradicts that limm→∞dm >0.

Thus, limn→∞d_n= 0. From (F3), there exists 0< r <1 such that

n→∞lim(d_n)^rF(d_n) = 0.

By (3.7), we get for all n∈N

(dn)^rF(d_n)−(dn)^rF(d₀)≤(dn)^rn(σ−τ(d−pn))≤0. (3.8) By letting n→ ∞ in (3.8), we obtain

n→∞lim n(d_n)^r= 0

This implies that there exists n1 ∈N such that n(dn)^r ≤ 1, or, dn ≤ ¹

n^1/r, for all n > n1. Rest of the proof can be completed as in Theorem 2.8.

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Following the arguments in the proof of Theorem 3.1 and takingF ∈F_∗, we obtain the following result.

Theorem 3.2. Let (X, d) be a complete metric space and α, η : X × X → R+ be two functions. Let T :X →C(X) andF ∈F_∗ satisfy all conditions of Theorem3.1. Then T has a fixed point inX.

By taking a₁= 1 and a₂ =a₃=a₄ =a₅ = 0 in Theorems 3.1 and 3.2 respectively, we get the following.

Corollary 3.3. Let (X, d) be a complete metric space and α, η : X × X → R+ be two functions. Let T :X →K(X) and F ∈F fulfill the following assertions:

3. there exist z0 ∈ X and y0 ∈ Tz0 such thatα(z0, y0)≥η(z0, y0);

and for any z∈ X with D(z,Tz)>0, there existsy∈ F_σ^z with α(z, y)≥η(z, y) satisfying τ(d(z, y)) +F(D(y,Ty))≤ F(d(z, y)).

By takinga₁ =a₂ =a₃ = 0 anda₄ =a₅= 1/2 in Theorems 3.1 and 3.2 respectively, we get the following results forF-contraction of Chatterjea type.

4. there exist σ >0 and a function τ : (0,∞)→(σ,∞) such that lim

t→s⁺infτ(t)> σ, for alls≥0,

and for any z∈ X with D(z,Tz)>0, there existsy∈ F_σ^z with α(z, y)≥η(z, y) satisfying τ(d(z, y)) +F(D(y,Ty))≤ F

D(z,Ty) +D(y,Tz) 2

. ThenT has a fixed point in X.

If we choose a₄ = a₅ = 0 in Theorems 3.1 and 3.2 respectively, we obtain the following results for F-contraction of Reich-type.

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and for any z∈ X with D(z,Tz)>0, there existsy∈ F_σ^z with α(z, y)≥η(z, y) satisfying τ(d(z, y)) +F(D(y,Ty))≤ F(a₁d(z, y) +a2D(z,Tz) +a3D(y,Ty)), where a1, a2, a3∈[0,+∞) such that a1+a2+a3= 1 and a3 6= 1.

Corollary 3.8. Let (X, d) be a complete metric space and α, η : X × X → R+ be two functions. Let T :X →C(X) andF ∈F∗ satisfy all conditions of Corollary3.7. Then T has a fixed point in X.

As an application of Theorems 3.1 and 3.2, we obtain the following.

Theorem 3.9. Let (X, d) be a complete metric space and α, η : X × X → R+ be two functions. Let T :X →K(X) and F ∈F fulfill the conditions(1) and (3) of Theorem 3.1 and the following assertions:

1. T is α-η continuous mapping;

2. there exists a functionτ : (0,∞)→(0,∞) such that

t→slim⁺infτ(t)>0, for all s≥0, and for anyy, z ∈ X withα(z, y)≥η(z, y) and H(Tz,Ty)>0 satisfying

τ(d(z, y)) +F(H(Tz,Ty))≤ F(a1d(z, y) +a2D(z,Tz) +a3D(y,Ty) +a₄D(z,Ty) +a₅D(y,Tz)),

wherea₁, a₂, a₃, a₄, a₅ ∈[0,+∞) such that a₁+a₂+a₃+ 2a₄= 1 and a₃6= 1.

Then T has a fixed point inX.

Proof. By Lemma 2.3, we have T is α-η-lower semi continuous mapping. Also, for z∈ X and y∈ F_σ^z with D(z,Tz)>0, we have

F(D(y,Ty))≤ F(H(Tz,Ty))≤ F(a₁d(z, y) +a₂D(z,Tz) +a₃D(y,Ty) +a₄D(z,Ty) +a₅D(y,Tz))−τ(d(z, y)).

Thus, all conditions of Theorem 3.1 are satisfied. Hence T has a fixed point.

By similar arguments of Theorem 3.9 and using Theorem 3.2, we state the following theorem.

Theorem 3.10. Let (X, d) be a complete metric space and α, η : X × X → R+ be two functions. Let T :X →C(X) andF ∈F∗ satisfy all conditions of Theorem3.9. Then T has a fixed point inX.

Theorem 3.11. Let(X, d)be a complete metric space, T :X →K(X)and F ∈F. If there existσ >0and a functionτ : (0,∞)→(σ,∞) such that

lim

t→s⁺infτ(t)> σ, for all s≥0, and for any z∈ X with D(z,Tz)>0, there existsy∈ F_σ^z satisfying

τ(d(z, y)) +F(D(y,Ty))≤ F(a₁d(z, y) +a₂D(z,Tz) +a₃D(y,Ty) +a₄D(z,Ty) +a₅D(y,Tz)),

where a1, a2, a3, a4, a5 ∈[0,+∞) such that a1+a2+a3+ 2a4 = 1 and a3 6= 1, then T has a fixed point in X providedz→D(z,Tz) is lower semi-continuous.

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Proof. Define α(z, y) = d(z, y) = η(z, y) for all z, y∈ X. Then by using Remark 2.7 and Theorem 3.1, we get the required result.

Theorem 3.12. Let (X, d) be a complete metric space, T :X →C(X) and F ∈F_∗ satisfy all assertions of Theorem 3.11. Then T has a fixed point in X.

Proof. Define α(z, y) = d(z, y) = η(z, y) for all z, y∈ X. Then by using Remark 2.7 and Theorem 3.2, we get the required result.

By takinga1= 1 anda2=a3 =a4=a5 = 0 in Theorems 3.11 and 3.12, we get the following corollaries.

Corollary 3.13 (Theorem 11 of [6]). Let (X, d) be a complete metric space,T :X →K(X) and F ∈F. If there exist σ >0 and a function τ : (0,∞)→(σ,∞) such that

lim

τ(d(z, y)) +F(D(y,Ty))≤ F(d(z, y)), thenT has a fixed point in X providedz→D(z,Tz) is lower semi-continuous.

Corollary 3.14 (Theorem 10 of [6]). Let (X, d) be a complete metric space, T : X → C(X) and F ∈ F_∗ satisfy all assertions of Corollary 3.13. Then T has a fixed point in X.

By takinga1 =a2=a3 = 0 anda4 =a5= 1/2 in Theorems 3.11 and 3.12, we get the following.

Corollary 3.15. Let (X, d) be a complete metric space, T :X → K(X) and F ∈ F. If there exist σ > 0 and a functionτ : (0,∞)→(σ,∞) such that

lim

τ(d(z, y)) +F(D(y,Ty))≤ F

D(z,Ty) +D(y,Tz) 2

, thenT has a fixed point in X providedz→D(z,Tz) is lower semi-continuous.

Corollary 3.16. Let (X, d) be a complete metric space, T :X →C(X) and F ∈F_∗ satisfy all assertions of Corollary 3.15. Then T has a fixed point in X.

By choosinga₄ =a₅= 0 in Theorems 3.11 and 3.12, we get the following.

Corollary 3.17. Let (X, d) be a complete metric space, T :X → K(X) and F ∈ F. If there exist σ > 0 and a functionτ : (0,∞)→(σ,∞) such that

lim

τ(d(z, y)) +F(D(y,Ty))≤ F(a1d(z, y) +a2D(z,Tz) +a3D(y,Ty)),

where a1, a2, a3 ∈ [0,+∞) such that a1+a2+a3 = 1 and a3 6= 1, then T has a fixed point in X provided z→D(z,Tz) is lower semi-continuous.

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Corollary 3.18. Let (X, d) be a complete metric space, T :X →C(X) and F ∈F_∗ satisfy all assertions of Corollary 3.17. Then T has a fixed point in X.

Remark 3.19. Corollary 3.13 is a generalization of Theorem 2.3 of [26]. In fact, if τ is a constant, then T is a multivaluedF-contraction and every multivaluedF-contraction is multivalued nonexpansive and every multivalued nonexpansive map is upper semi-continuous, thenT is upper semi-continuous. Therefore, the functionz →D(z,Tz) is lower semi-continuous. On the other hand for any z∈ X withD(z,Tz)>0 and y∈ F_σ^z, we have

τ(d(z, y)) +F(D(y,Ty))≤τ(d(z, y)) +F(H(Tz,Ty))≤ F(d(z, y)).

HenceT satisfies all conditions of Corollary 3.13. Similarly, Corollary 3.14 generalizes Theorem 2.5 of [26].

Remark 3.20. If we take T, a single self-mapping on X, Theorems 3.11 and 3.12 reduce to Theorem 1 of [30].

4. Fixed point results in partially ordered metric space

Let (X, d,) be a partially ordered metric space and T : X → 2^X be a multivalued mapping. For A, B ∈ 2^X, A B implies that a b for all a ∈ A and b ∈ B. We say that T is monotone increasing, if Ty Tz, for all y, z ∈ X, for which y z. There are many applications in differential and integral equations of monotone mappings in ordered metric spaces (see [2, 7, 16, 17] and references therein). In this section, from Sections 2 and 3, we derive the following new results in partially ordered metric spaces.

Theorem 4.1. Let (X, d,) be a complete partially ordered metric space, T : X → K(X), F ∈ F and G ∈G fulfill the following assertions:

1. if for anyz∈ X with D(z,Tz)>0, there existsy∈ F_σ^z with zy satisfying

G(D(z,Tz), D(y,Ty), D(z,Ty), D(y,Tz)) +F(D(y,Ty))≤ F(d(z, y));

2. T is monotone increasing;

3. there exist z0 ∈ X and y0 ∈ Tz0 such thatz0 y0;

4. for givenz∈X and sequence {z_n} with zn→z as n→ ∞ andznzn+1 for alln∈N, we have

n→∞lim infD(z_n,Tz_n)≥D(z,Tz), thenT has a fixed point in X providedσ < τ.

Proof. Define α, η:X × X →[0,∞) by α(z, y) =

2 zy,

0 otherwise, η(z, y) =

1 zy, 0 otherwise,

then for z, y ∈ X with z y, α(y, z) ≥ η(y, z) implies u v for all u ∈ Tz and v ∈ Ty. Hence α(u, v) = 2 > 1 = η(u, v), for all u ∈ Tz and v ∈ Ty and α(u, v) = η(u, v) = 0 otherwise. This shows thatT is generalized α∗-admissible mapping with respect toη. Thus, all the conditions of Theorem 2.8 are satisfied and T has a fixed point.

By similar arguments as in Theorem 4.1, we state the following.

Theorem 4.2. Let (X, d,) be a complete partially ordered metric space, T : X → C(X), F ∈ F_∗ and G ∈G fulfill all conditions of Theorem 4.1. Then T has a fixed point in X providedσ < τ.

Theorem 4.3. Let (X, d,) be a complete partially ordered metric space, T : X → K(X), F ∈ F and G ∈G fulfill the conditions (2) and(3) of Theorem 4.1 and the following assertions:

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1. If for anyz, y∈ X withzy andH(Tz,Ty)>0 satisfying

G(D(z,Tz), D(y,Ty), D(z,Ty), D(y,Tz)) +F(H(Tz,Ty))≤ F(d(z, y));

2. for given z ∈ X and sequence {z_n} with zn → z as n → ∞ and zn zn+1 for all n ∈ N, we have Tz_n→ Tz,

thenT has a fixed point in X.

Theorem 4.4. Let (X, d,) be a complete partially ordered metric space, T : X → C(X), F ∈ F_∗ and G ∈G fulfill all conditions of Theorem 4.3. Then T has a fixed point in X.

By takingG =G_L, as in Corollary 2.10, Theorems 4.1–4.4 reduce to the following.

Corollary 4.5. Let (X, d,) be a complete partially ordered metric space, T : X → K(X) and F ∈ F satisfy conditions(2)-(4) of Theorem4.1 and if for any z∈ X with D(z,Tz) >0, there exists y∈ F_σ^z with zy satisfying

τ+F(D(y,Ty))≤ F(d(z, y)), thenT has a fixed point in X providedσ < τ.

Corollary 4.6. Let (X, d,) be a complete partially ordered metric space, T : X → C(X) and F ∈ F_∗ satisfy all conditions of Corollary4.5. Then T has a fixed point in X provided σ < τ.

Corollary 4.7. Let(X, d,)be a complete partially ordered metric space, T :X →K(X) andF ∈F fulfill conditions (2)-(4) of Theorem 4.1 and if for any z∈ X with zy andH(Tz,Ty)>0 we have

τ +F(H(Tz,Ty))≤ F(d(z, y)), thenT has a fixed point in X.

Corollary 4.8. Let (X, d,) be a complete partially ordered metric space, T : X → C(X) and F ∈ F_∗ satisfy all conditions of Corollary4.7. Then T has a fixed point in X provided σ < τ.

Theorem 4.9. Let (X, d) be a complete metric space, T : X → K(X) and F ∈ F fulfill the following assertions:

1. T is monotone increasing;

2. there exist z₀ ∈ X and y₀ ∈ Tz₀ such thatz₀ y₀;

3. for givenz∈X and sequence {z_n} with z_n→z as n→ ∞ andz_nz_n+1 for alln∈N we have

n→∞lim infD(z_n,Tz_n)≥D(z,Tz);

and for any z∈ X with D(z,Tz)>0, there existsy∈ F_σ^z with zy satisfying τ(d(z, y)) +F(D(y,Ty))≤ F(a₁d(z, y) +a₂D(z,Tz) +a₃D(y,Ty)

+a₄D(z,Ty) +a₅D(y,Tz)), where a1, a2, a3, a4, a5∈[0,+∞) such that a1+a2+a3+ 2a4= 1 and a3 6= 1.

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Theorem 4.10. Let (X, d,) be a complete partially ordered metric space, T : X → C(X) and F ∈ F_∗ fulfill all conditions of Theorem 4.9. Then T has a fixed point in X provided σ < τ.

Theorem 4.11. Let(X, d,)be a complete partially ordered metric space,T :X →K(X)andF ∈Ffulfill the conditions(1) and (2)of Theorem 4.9and the following assertions:

1. for given z ∈ X and sequence {z_n} with zn → z as n → ∞ and zn zn+1 for all n ∈ N, we have Tzn→ Tz;

t→slim⁺infτ(t)> σ, for alls≥0, and for any z, y∈ X with zy and H(Tz,Ty)>0, satisfying

τ(d(z, y)) +F(H(Tz,Ty))≤ F(a1d(z, y) +a2D(z,Tz) +a3D(y,Ty) +a4D(z,Ty) +a5D(y,Tz)),

where a₁, a₂, a₃, a₄, a₅∈[0,+∞) such that a₁+a₂+a₃+ 2a₄= 1 and a₃ 6= 1.

Theorem 4.12. Let (X, d,) be a complete partially ordered metric space, T : X → C(X) and F ∈ F_∗ fulfill all conditions of Theorem 4.11. Then T has a fixed point in X providedσ < τ.

5. Suzuki-Wardowski type fixed point results

In this section we establish certain fixed point results for Suzuki-Wardowski type multivaluedF-contractions.

Theorem 5.1. Let (X, d) be a complete metric space, T : X → K(X) and F ∈ F. If for z, y ∈ X with

1

2D(z,Tz)≤d(z, y) andD(z,Tz)>0, we have

τ+F(D(y,Ty))≤ F(d(z, y)), (5.1) thenT has a fixed point in X providedz→D(z,Tz) is lower semi-continuous.

Proof. Suppose that G =G_L as in Corollary 2.10. Let z∈ X withD(z,Tz)>0 and y ∈ F_σ^z,σ < τ. Then y∈ Tz, therefore we have ¹₂D(z,Tz)≤D(z,Tz)≤d(z, y). So, by using (5.1), we get

G(D(z,Tz), D(y,Ty), D(z,Ty), D(y,Tz)) +F(D(y,Ty)) =τ +F(D(y,Ty))

≤ F(d(z, y)).

Thus, all conditions of Theorem 2.18 hold and T has a fixed point.

Theorem 5.2. Let (X, d) be a complete metric space, T : X →C(X) and F ∈ F∗. If for z, y ∈ X with

1

2D(z,Tz)≤d(z, y) and D(z,Tz)>0, we have

τ+F(D(y,Ty))≤ F(d(z, y)),

thenT has a fixed point in X providedz→D(z,Tz) is lower semi-continuous.

Proof. By takingG=G_L as in Corollary 2.10 and by using Theorem 2.19, we get the required result.

Theorem 5.3. Let (X, d) be a complete metric space, T : X → K(X) and F ∈ F. If for z, y ∈ X with

1

2D(z,Tz)≤d(z, y) and H(Tz,Ty)>0, we have

τ +F(H(Tz,Ty))≤ F(d(z, y)), thenT has a fixed point in X.